As an AI language model, I cannot perform quantum chemical calculations directly. However, I can guide you through the process and provide some general information about the vibrational frequencies of carbon-oxygen bonds.To calculate the vibrational frequency of the carbon-oxygen double bond in ethylene oxide, you would need to perform quantum chemical calculations using software like Gaussian, ORCA, or GAMESS. These programs use various methods, such as Density Functional Theory DFT or Hartree-Fock HF , to calculate the electronic structure and vibrational frequencies of molecules.Once you have the vibrational frequencies, you can compare them to the corresponding frequency for a single carbon-oxygen bond. In general, the vibrational frequency of a double bond is higher than that of a single bond. This is because the double bond is stronger and stiffer, which results in a higher force constant and, consequently, a higher vibrational frequency.The infrared IR spectra of the two bonds will also differ due to the differences in their vibrational frequencies. In the IR spectra, the vibrational frequencies appear as peaks, and the peak positions are related to the bond strengths. The peak for the carbon-oxygen double bond will appear at a higher wavenumber cm than the peak for the single bond, reflecting the higher vibrational frequency of the double bond.To summarize, the vibrational frequency of the carbon-oxygen double bond in ethylene oxide will be higher than that of a single carbon-oxygen bond, and this difference will be reflected in their respective IR spectra. To obtain the exact frequencies and spectra, you would need to perform quantum chemical calculations using appropriate software.